Letters

Part III
Richard Wein is persuaded that I have committed a great absurdity in insisting that Nilsson and Pelger's paper contains nothing corresponding to the requisite random variations that are called for in Darwinian theory. He is quite mistaken. What is at issue is whether changes in a population from one generation to another arise as the expression of an underlying stochastic process. It is, of course, the only point at issue. If the dynamics of change are not essentially stochastic, the underlying theory is not Darwinian. I take it that this is by definition what a Darwinian theory requires. In their work, Nilsson and Pelger do specify an initial probability distribution over a sample population. They never provide the requisite probability transition system. Their dynamical model is entirely deterministic.
Nilsson and Pelger's results are presented in two stages. In the first, Nilsson and Pelger investigate the fate of a single light sensitive cell. Nilsson and Pelger assume that changes to their initial patch are governed by the function

1) f(a)n

subject to the boundary condition that for a = 1.01, f(a)n = 80, 129, 540, whence n = 1829.
1), of course, is simply a stripped-down formula for the generation of compound interest. 1) determines an action f --> f --> ... --> f, such that the transition probability Pr(fifi+1) = 1 for any two actions of the function. The point is again obvious, indeed, trivial. No Darwinian assumptions are at work; no stochastic features of any sort.
In the second stage of their paper, Nilsson and Pelger discount a by means of Falconer's short term response statistic R. Short term, note. Falconer's response statistic is not meant to model the long term behavior of a sample population. The discounted value of a now stands for the mean value MV of visual acuity in every generation.

Part IIIa
Let us now normalize Nilsson and Pelger's dynamic equation by setting coefficients of heredity and selection to 1. Neither selection nor heredity address the source of dynamical change. Their dynamical theory is again expressed as a function

2) f(MVINITIAL)n.

But since MVINITIAL = 1.01, when coefficients of heredity and selection are 1, 2) is nothing more than a restatement of 1), subject, in fact, to precisely the same boundary condition: f(MVINITIAL)n = 80, 129, 540, whence n = 1829 again.
It follows again that the transition probability for each fifi+1 = 1. No surprise, this.
To regard this as a Darwinian model, or to imagine that 2) specifies a probability transition system, is simply a mistake.
What might reasonable transition probabilities look like? I have no idea, of course, but there is absolutely no reason to suppose that the MV will rise monotonically throughout the course of 330,000 generations. It may well decline, leading to negative selection and a net reduction of population, or it may oscillate persistently around the initial mean. But one thing is clear. Nilsson and Pelger's model requires that an emerging eye undergo 1829 consecutive positive one percent changes. If each one percent step corresponds to a mutation, I would conservatively estimate that the chances of observing 1829 consecutive positive mutations in precisely the same genetic locus is effectively zero. So would everyone else.

Part IV
In my Commentary article I observed that Nilsson and Pelger provided no calculations justifying their central claim, namely that 1829 one percent steps are sufficient to transform a light sensitive patch into a camera eye. The list that Mr. Curtis offers is of scant help in this regard. More specifically,
1 It is incomplete
2 It is largely incomprehensible. What, for example, does "corneal width (curve) 46.5 46.50" mean?
3 It is irrelevant inasmuch as it was not included in Nilsson and Pelger's original paper nor in their notes.
And, finally
4 It is absurd, inasmuch as a list is not a calculation. In my essay, I asked how Nilsson and Pelger's numbers were derived? They do not say and neither does Mr. Curtis. Suppose, for example, that Nilsson and Pelger's original light sensitive patch were to increase in length, and length alone, by 1829 one percent steps. Would that result in a structure similar to the one they derive, or would it be different? In either case, how would one know, without a specification of overall morphological change by means of a single derived unit of morphological change amalgamating all the dimensions of change? A very rich literature now exists dealing with the metric structure of complex three-dimensional biological objects. References are available on line under my name at the Discovery Institute web site.
One final point. I have never accused Nilsson and Pelger of fraud. I consider their paper absurd, but that is another matter entirely. The fraud in question involves the misrepresentation of their work, chiefly but not only by Richard Dawkins.

David Berlinski writes: "What is at issue is whether changes in a population from one generation to another arise as the expression of an underlying stochastic process. It is, of course, the only point at issue."

On the contrary, it is not at all the point at issue! Nilsson and Pelger take for granted the fact that such changes can arise. The justification for that assumption is outside the scope of their paper. The issue they address is whether such changes can accumulate to produce an eye, and how long such a process is likely to take.

Berlinski appears to believe that a stochastic process cannot be represented by a deterministic model. If so, he is clearly wrong. Deterministic algorithms are often used to model stochastic processes. Does a casino manager use cards, dice or other randomizers in estimating his future takings? I doubt it. Due to the large number of random events, he makes the approximating assumption that his takings will be in accordance with the statistical expectation, which he calculates by means of a purely deterministic algorithm. Similarly, Nilsson and Pelger make the approximating assumption that the quantitative change in each generation will be the statistically expectated value estimated by Falconer's formula.

Berlinski is too concerned with whether Nilsson and Pelger's model can be labelled "Darwinian". I'm not even sure what he means by this. If he means that their algorithm is not Darwinian, then I agree. But just as one can use a non-stochastic algorithm to model a stochastic process, one can use a non-Darwinian algorithm to model a Darwinian process. Berlinski is chasing a red herring. The real issue is not how we label the model, but whether the assumptions and approximations it makes are justifiable.

His failure to appreciate that a model is an abstraction leads Berlinski into further blunders: "...there is absolutely no reason to suppose that the MV will rise monotonically throughout the course of 330,000 generations." Indeed, but the model does not require that it do so. It merely assumes that it will do so for the purposes of approximation. All the model reqires is that the mean value rises on average by roughly the calculated amount. Sometimes it may fall, other times it may rise faster and catch up.

[Regarding my allegation of argument by innuendo, I will respond to Berlinski in the column where I originally made that allegation (titled "James Downard").]